About the Project
18 Orthogonal PolynomialsGeneral Orthogonal Polynomials

§18.2 General Orthogonal Polynomials

  1. §18.2(i) Definition
  2. §18.2(ii) x-Difference Operators
  3. §18.2(iii) Standardization and Related Constants
  4. §18.2(iv) Recurrence Relations
  5. §18.2(v) Christoffel–Darboux Formula
  6. §18.2(vi) Zeros
  7. §18.2(vii) Quadratic Transformations
  8. §18.2(viii) Uniqueness of Orthogonality Measure and Completeness
  9. §18.2(ix) Moments
  10. §18.2(x) Orthogonal Polynomials and Continued Fractions
  11. §18.2(xi) Some Special Classes of General Orthogonal Polynomials
  12. §18.2(xii) Other Special Constructions Involving General OP’s

§18.2(i) Definition

Orthogonality on Intervals

Let (a,b) be a finite or infinite open interval in . A system (or set) of polynomials {pn(x)}, n=0,1,2,, where pn(x) has degree n as in §18.1(i), is said to be orthogonal on (a,b) with respect to the weight function w(x) (0) if

18.2.1 abpn(x)pm(x)w(x)dx=0,

Here w(x) is continuous or piecewise continuous or integrable such that

18.2.1_5 0<abw(x)dx<,

It is assumed throughout this chapter that for each polynomial pn(x) that is orthogonal on an open interval (a,b) the variable x is confined to the closure of (a,b) unless indicated otherwise. (However, under appropriate conditions almost all equations given in the chapter can be continued analytically to various complex values of the variables.)

Orthogonality on Countable Sets

Let X be a finite set of distinct points on , or a countable infinite set of distinct points on , and wx, xX, be a set of positive constants. Then a system of polynomials {pn(x)}, n=0,1,2,, is said to be orthogonal on X with respect to the weights wx if

18.2.2 xXpn(x)pm(x)wx=0,

when X is infinite, or

18.2.3 xXpn(x)pm(x)wx=0,

when X is a finite set of N+1 distinct points. In the former case we also require

18.2.4 xX|x|nwx<,

whereas in the latter case the system {pn(x)} is finite: n=0,1,,N.

Orthogonality on General Sets

More generally than (18.2.1)–(18.2.3), w(x)dx may be replaced in (18.2.1) by dμ(x), where the measure μ is the Lebesgue–Stieltjes measure μα corresponding to a bounded nondecreasing function α on the closure of (a,b) with an infinite number of points of increase, and such that ab|x|ndμ(x)< for all n. See §1.4(v), McDonald and Weiss (1999, Chapters 3, 4) and Szegő (1975, §1.4). Then

18.2.4_5 abpn(x)pm(x)dμ(x)=0,

§18.2(ii) x-Difference Operators

If the orthogonality discrete set X is {0,1,,N} or {0,1,2,}, then the role of the differentiation operator d/dx in the case of classical OP’s (§18.3) is played by Δx, the forward-difference operator, or by x, the backward-difference operator; compare §18.1(i). This happens, for example, with the Hahn class OP’s (§18.20(i)).

If the orthogonality interval is (,) or (0,), then the role of d/dx can be played by δx, the central-difference operator in the imaginary direction (§18.1(i)). This happens, for example, with the continuous Hahn polynomials and Meixner–Pollaczek polynomials (§18.20(i)).

§18.2(iii) Standardization and Related Constants

The orthogonality relations (18.2.1)–(18.2.3) each determine the polynomials pn(x) uniquely up to constant factors, which may be fixed by suitable standardizations.


Throughout this chapter we will use constants hn and kn, and variants of these, related to OP’s pn(x).

The hn are defined as:

18.2.5 hn =ab(pn(x))2w(x)dx
or hn =xX(pn(x))2wx
or hn =ab(pn(x))2dμ(x).


18.2.5_5 abpn(x)pm(x)w(x)dx=hnδn,m,

and similar extensions for (18.2.4_5) and (18.2.2).

The constants h~n, kn, k~n and k~~n are defined as:

18.2.6 h~n =abx(pn(x))2w(x)dx
or h~n =xXx(pn(x))2wx
or h~n =abx(pn(x))2dμ(x),


18.2.7 pn(x)=knxn+k~nxn1+k~~nxn2+,

where k~0=0 and k~~n=0 for n=0,1.


The classical orthogonal polynomials are defined with:

(i) the traditional OP standardizations of Table 18.3.1, where each is defined in terms of the above constants.

Two, more specialized, standardizations are:

(ii) monic OP’s: kn=1.

(iii) orthonormal OP’s: hn=1 (and usually, but not always, kn>0);

The constant function p0(x) will often, but not always, be identically 1 (see, for example, (18.2.11_8)), p1(x)=0 in all cases, by convention, as indicated in §18.1(i).

§18.2(iv) Recurrence Relations

As in §18.1(i) we assume that p1(x)0.

First Form

18.2.8 pn+1(x)=(Anx+Bn)pn(x)Cnpn1(x),

Here An, Bn (n0), and Cn (n1) are real constants. Then

18.2.9 An =kn+1kn,
Bn =(k~n+1kn+1k~nkn)An=h~nhnAn,
Cn =Ank~~n+Bnk~nk~~n+1kn1=AnAn1hnhn1.

Hence An1AnCn=An2hn/hn1 (n1), so

18.2.9_5 An1AnCn>0,

The OP’s are orthonormal iff Cn=An/An1 (n1) and h0=1. The OP’s are monic iff An=1 (n0) and k0=1.

Second Form

18.2.10 xpn(x)=anpn+1(x)+bnpn(x)+cnpn1(x),

Here an, bn (n0), cn (n1) are real constants. Then

18.2.11 an =knkn+1,
bn =k~nknk~n+1kn+1=h~nhn,
cn =k~~nank~~n+1bnk~nkn1=an1hnhn1.


18.2.11_1 j=0nbj=k~n+1kn+1.

Furthermore, an1cn=an12hn/hn1 (n1), so

18.2.11_2 an1cn>0,

The OP’s are orthonormal iff cn=an1 (n1) and h0=1. The OP’s are monic iff an=1 (n0) and k0=1.

The coefficients An,Bn,Cn in the first form and an,bn,cn in the second form are related by

18.2.11_3 an =An1,bn=An1Bn,cn=An1Cn;
An =an1,Bn=an1bn,Cn=an1cn.

Monic and Orthonormal Forms

Assume that the pn(x) are monic, so An=1=an. Then, with

18.2.11_4 αn bn=Bn,
βn cn=Cn=hn/hn1,

the monic recurrence relations (18.2.8) and (18.2.10) take the form

18.2.11_5 xpn(x) =pn+1(x)+αnpn(x)+βnpn1(x),
p1(x) =xα0,
p0(x) =1.

See also (3.5.30). Note that

18.2.11_6 βn>0,

In terms of the monic OP’s pn define the orthonormal OP’s qn by

18.2.11_7 qn(x)=pn(x)/hn,

Then, with the coefficients (18.2.11_4) associated with the monic OP’s pn, the orthonormal recurrence relation for qn takes the form

18.2.11_8 xqn(x) =βn+1qn+1(x)+αnqn(x)+βnqn1(x),
q1(x) =(xα0)/h0β1,
q0(x) =1/h0,

with h0 still being associated with the monic p0(x)=1.

The monic and orthonormal OP’s, and their determination via recursion, are more fully discussed in §§3.5(v) and 3.5(vi), where modified recursion coefficients are listed for the classical OP’s in their monic and orthonormal forms.


If polynomials pn(x) are generated by recurrence relation (18.2.8) under assumption of inequality (18.2.9_5) (or similarly for the other three forms) then the pn(x) are orthogonal by Favard’s theorem, see §18.2(viii), in that the existence of a bounded non-decreasing function α(x) on (a,b) yielding the orthogonality realtion (18.2.4_5) is guaranteed.

If the polynomials pn(x) (n=0,1,,N) are orthogonal on a finite set X of N+1 distinct points as in (18.2.3), then the polynomial pN+1(x) of degree N+1, up to a constant factor defined by (18.2.8) or (18.2.10), vanishes on X.

The recurrence relations (18.2.10) can be equivalently written as

18.2.11_9 (b0a00c1b1a1c20)(p0(x)p1(x))=x(p0(x)p1(x)).

The matrix on the left-hand side is an (infinite tridiagonal) Jacobi matrix. This matrix is symmetric iff cn=an1 (n1).

§18.2(v) Christoffel–Darboux Formula

With notation (18.2.4_5), (18.2.5), (18.2.7)

18.2.12 Kn(x,y)=0np(x)p(y)h=knhnkn+1pn+1(x)pn(y)pn(x)pn+1(y)xy,

Kernel property

18.2.12_5 abf(y)Kn(x,y)dμ(y)={f(x),fSpan(p0,p1,,pn),0,abf(x)p(x)dμ(x)=0 (=0,1,,n).

Confluent Form

18.2.13 Kn(x,x)==0n(p(x))2h=knhnkn+1(pn+1(x)pn(x)pn(x)pn+1(x)).

Kernel Polynomials

Assume y(a,b) in (18.2.12). Then the kernel polynomials

18.2.14 qn(x)=qn(x;y)Kn(x,y)==0np(x)p(y)h

are OP’s with orthogonality relation

18.2.15 abqn(x)qm(x)|yx|dμ(x)=0,

Between the systems {pn(x)} and {qn(x)} there are the contiguous relations

18.2.16 qn(x)qn1(x) =pn(y)hnpn(x),
18.2.17 pn(y)pn+1(x)pn+1(y)pn(x) =hnkn+1kn(xy)qn(x).

§18.2(vi) Zeros

All n zeros of an OP pn(x) are simple, and they are located in the interval of orthogonality (a,b). The zeros of pn(x) and pn+1(x) separate each other, and if m<n then between any two zeros of pm(x) there is at least one zero of pn(x).

For illustrations of these properties see Figures

For usage of the zeros of an OP in Gauss quadrature see §3.5(v). When the Jacobi matrix in (18.2.11_9) is truncated to an n×n matrix

18.2.18 𝐉n=(b0a00c1b1a1c2an20cn1bn1)

then the zeros of pn(x) are the eigenvalues of 𝐉n (see also §3.5(vi)).


Let x1,,xn be the zeros of the OP pn, so

18.2.19 pn(x)=knj=1n(xxj).

The discriminant of pn is defined by

18.2.20 Disc(pn)=kn2n21i<jn(xixj)2.

See Ismail (2009, §3.4) for another expression of the discriminant in the case of a general OP.

§18.2(vii) Quadratic Transformations

For OP’s {pn(x)} on with respect to an even weight function w(x) we have

18.2.21 pn(x)=(1)npn(x),

so we can put

18.2.22 rn(x2) p2n(x),
sn(x2) x1p2n+1(x),
v(x2) w(x).

Then {rn(x)} are OP’s on (0,) with respect to weight function x12v(x) and {sn(x)} are OP’s on (0,) with respect to weight function x12v(x).

As a slight variant let {pn(x)} be OP’s with respect to an even weight function w(x) on (1,1). Then (18.2.21) still holds and we can put

18.2.23 rn(2x21) p2n(x),
sn(2x21) x1p2n+1(x),
v(2x21) w(x).

Then {rn(x)} are OP’s on (1,1) with respect to weight function (1+x)12v(x) and {sn(x)} are OP’s on (1,1) with respect to weight function (1+x)12v(x).

See Chihara (1978, Ch. I, §8).

§18.2(viii) Uniqueness of Orthogonality Measure and Completeness

If a system of polynomials {pn(x)} satisfies any of the formula pairs (recurrence relation and coefficient inequality) (18.2.8), (18.2.9_5) or (18.2.10), (18.2.11_2) or (18.2.11_5), (18.2.11_6) or (18.2.11_8), (18.2.11_6) then {pn(x)} is orthogonal with respect to some positive measure on (Favard’s theorem). The measure is not necessarily absolutely continuous (i.e., of the form w(x)dx) nor is it necessarily unique, up to a positive constant factor. However, if OP’s have an orthogonality relation on a bounded interval, then their orthogonality measure is unique, up to a positive constant factor.

A system {pn(x)} of OP’s satisfying (18.2.1) and (18.2.5) is complete if each f(x) in the Hilbert space Lw2((a,b)) can be approximated in Hilbert norm by finite sums nλnpn(x). For such a system, functions fLw2((a,b)) and sequences {λn} (n=0,1,2,) satisfying n=0hn|λn|2< can be related to each other in a similar way as was done for Fourier series in (1.8.1) and (1.8.2):

18.2.24 λn=hn1abf(x)pn(x)w(x)dx

if and only if

18.2.25 f(x)=n=0λnpn(x)

(convergence in Lw2((a,b))). A system of OP’s with unique orthogonality measure is always complete, see Shohat and Tamarkin (1970, Theorem 2.14). In particular, a system of OP’s on a bounded interval is always complete.

§18.2(ix) Moments

The moments for an orthogonality measure dμ(x) are the numbers

18.2.26 μn=abxndμ(x),

The Hankel determinant Δn of order n is defined by Δ0=1 and

18.2.27 Δn=|μ0μ1μn1μ1μ2μnμn1μnμ2n2|,

Also define determinants Δn by Δ0=0, Δ1=μ1 and

18.2.28 Δn=|μ0μ1μn2μnμ1μ2μn1μn+1μn1μnμ2n3μ2n1|,

The monic OP’s pn(x) with respect to the measure dμ(x) can be expressed in terms of the moments by

18.2.29 pn(x)=1Δn|μ0μ1μnμ1μ2μn+1μn1μnμ2n11xxn|,

The recurrence coefficients αn and βn in (18.2.11_5) can be expressed in terms of the determinants (18.2.27) and (18.2.28) by

18.2.30 αn =Δn+1Δn+1ΔnΔn,
βn =Δn+1Δn1Δn2,

It is to be noted that, although formally correct, the results of (18.2.30) are of little utility for numerical work, as Hankel determinants are notoriously ill-conditioned. See Gautschi (2004, p. 54), and Golub and Meurant (2010, pp. 56, 57). Alternatives for numerical calculation of the recursion coefficients in terms of the moments are discussed in these references, and in §18.40(ii).

§18.2(x) Orthogonal Polynomials and Continued Fractions

In this subsection fix the recurrence coefficients αn (n=0,1,2,) and βn (n=1,2,) as in (18.2.11_5), with pn(x) the corresponding monic OP’s and with dμ(x), a and b as in the orthogonality relation (18.2.4_5). Define the first associated monic orthogonal polynomials pn(1)(x) as monic OP’s satisfying

18.2.31 p0(1)(x) =1,
p1(1)(x) =xα1,
xpn(1)(x) =pn+1(1)(x)+αn+1pn(1)(x)+βn+1pn1(1)(x),

where the first indicates that the indices of the recursion coefficients αn, βn of (18.2.31) have been incremented by 1, when compared to those of (18.2.11_5). More generally, §18.30 defines the recurrence relation of the cth associated monic OP by means of a similar shift by c in (18.2.11_5).

The OP’s pn(1)(x) may also be calculated from the original recursion (18.2.11_5), but with independent initial conditions for p0,p1:

18.2.32 p0(0)(x) =0,
p1(0)(x) =1,
xpn(0)(x) =pn+1(0)(x)+αnpn(0)(x)+βnpn1(0)(x),

resulting in pn(0)(x)=pn1(1)(x), by simple comparison of the two recursions. The pn(0)(x) are the monic corecursive orthogonal polynomials. These relationships are further explored in §§18.30(vi) and 18.30(vii).

The polynomials pn(1)(x) may be also be directly expressed in terms of the pn(x) of (18.2.11_5):

18.2.33 pn1(1)(z)=1μ0abpn(z)pn(x)zxdμ(x),
z\[a,b], n=1,2,,

with moment μ0 defined in (18.2.26).

Using the terminology of §1.12(ii), the n-th approximant of the continued fraction

18.2.34 1xα0β1xα1β2xα2

is given by

18.2.35 Fn(x)=1xα0β1xα1β2xα2βn1xαn1.


18.2.36 Fn(x)=pn1(1)(x)pn(x)=pn(0)(x)pn(x)=1μ0k=1nwkxxk,

where x1,x2,,xn are the zeros of pn(x) and

18.2.37 wk=abpn(x)(xxk)pn(xk)dμ(x),

are the Christoffel numbers, see also (3.5.18). Because of (18.2.36) the OP’s pn(x) are also called monic denominator polynomials and the OP’s pn1(1)(x), or, equivalently, the pn(0)(x), are called the monic numerator polynomials.

Assume that the interval [a,b] is bounded. Markov’s theorem states that

18.2.38 limnFn(z)=1μ0abdμ(x)zx,

See Chihara (1978, pp. 86–89), and, in slightly different notation, Ismail (2009, §§2.3, 2.6, 2.10), where it is assumed that μ0=1. See also the extended development of these ideas in §§18.30(vi), 18.30(vii), and in §18.40(ii) where they form the basis for one method of solving the classical moment problem.

§18.2(xi) Some Special Classes of General Orthogonal Polynomials

The Szegő Class 𝒢

This is the class of weight functions w on (1,1) such that, in addition to (18.2.1_5),

18.2.39 11|ln(w(x))|1x2dx<.

For OP’s pn(x) with weight function in the class 𝒢 there are asymptotic formulas as n, respectively for x outside [1,1] and for x[1,1], see Szegő (1975, Theorems 12.1.2, 12.1.4). Under further conditions on the weight function there is an equiconvergence theorem, see Szegő (1975, Theorem 13.1.2). This says roughly that the series (18.2.25) has the same pointwise convergence behavior as the same series with pn(x)=Tn(x), a Chebyshev polynomial of the first kind, see Table 18.3.1.

Generalizations of the Szegő Class

Nevai (1979, p.39) defined the class 𝒮 of orthogonality measures with support inside [1,1] such that the absolutely continuous part w(x)dx has w in the Szegő class 𝒢. For OP’s with orthogonality measure in 𝒮 Nevai (1979, pp. 148–150) generalized Szegő’s equiconvergence theorem. In further generalizations of the class 𝒮 discrete mass points xk outside [1,1] are allowed. If these xk satisfy k(|xk|1)1/2< then Szegő type asymptotics outside [1,1] can be given for the corresponding OP’s, see Simon (2011, Corollary 3.7.2 and following).

The Nevai class 𝐌(a,b)

The class 𝐌(a,b) (a>0, b), introduced by Nevai (1979, p.10), consists of all orthogonality measures dμ such that the coefficients βn and αn in the recurrence relation (18.2.11_8) for the corresponding orthonormal OP’s satisfy

18.2.40 limnβn =12a,
limnαn =b.

If dμ𝐌(a,b) then the interval [ba,b+a] is included in the support of dμ, and outside [ba,b+a] the measure dμ only has discrete mass points xk such that b±a are the only possible limit points of the sequence {xk}, see Máté et al. (1991, Theorem 10). Part of this theorem was already proved by Blumenthal (1898). Therefore this class is also called the Nevai–Blumenthal class.

§18.2(xii) Other Special Constructions Involving General OP’s

Poisson kernel

For OP’s pn with hn and orthogonality relation as in (18.2.5) and (18.2.5_5), the Poisson kernel is defined by

18.2.41 Pz(x,y)=n=0pn(x)pn(y)hnzn,

for x,y in the support of the orthogonality measure and z such that the series in (18.2.41) converges absolutely for all these x,y. Instances where the Poisson kernel is nonnegative are of special interest, see Ismail (2009, Theorem 4.7.12).

Degree lowering and raising differentiation formulas and structure relations

For a large class of OP’s pn there exist pairs of differentiation formulas

18.2.42 πn(x)pn(x)+An(x)pn(x) =λnpn1(x),
18.2.43 πn(x)pn(x)+Bn(x)pn(x) =μnpn+1(x),

see Ismail (2009, (3.2.3), (3.2.10)). If An(x) and Bn(x) are polynomials of degree independent of n, and moreover πn(x) is a polynomial π(x) independent of n then

18.2.44 π(x)pn(x)=j=nsn+tan,jpj(x)

for certain coefficients an,j with s,t independent of n. Then the OP’s are called semi-classical and (18.2.44) is called a structure relation.

Sheffer Polynomials

Polynomials pn(x) of degree n (n=0,1,2,) are called Sheffer polynomials if they are generated by a generating function of the form

18.2.45 f(t)exp(xu(t))=n=0cnpn(x)tnn!,

where f(t) and u(t) are formal power series in t, with f(0)=1, u(0)=0 and u(0)=1. Often a standardization cn=1 is taken. If v(s) is the formal power series such that v(u(t))=t then a property equivalent to (18.2.45) with cn=1 is that

18.2.46 v(Dx)pn(x)=npn1(x).

The operator Dx is a delta operator, i.e., Dx commutes with translation in the variable x and Dxx is a nonzero constant.

The generating functions (18.12.13), (18.12.15), (18.23.3), (18.23.4), (18.23.5) and (18.23.7) for Laguerre, Hermite, Krawtchouk, Meixner, Charlier and Meixner–Pollaczek polynomials, respectively, can be written in the form (18.2.45). In fact, these are the only OP’s which are Sheffer polynomials (with Krawtchouk polynomials being only a finite system)

The Bernoulli polynomials Bn(x) and Euler polynomials En(x) are examples of Sheffer polynomials which are not OP’s, see the generating functions (24.2.3) and (24.2.8). For other examples of Sheffer polynomials, not in DLMF, see Roman (1984).

For further details see Meixner (1934), Sheffer (1939), Rota et al. (1973) and Butzer and Koornwinder (2019).

Monotonic Weight Functions

For OP’s pn on [a,b] with weight function w(x) and orthogonality relation (18.2.5_5) assume that b< and w(x) is non-decreasing in the interval [a,b]. Then the functions w(x)pn(x) attain their maximum in [a,b] for x=b. See Szegő (1975, Theorem 7.2).

Equations (18.14.3_5) and (18.14.8), both for α=0, can be seen as special cases of this result for Jacobi and Laguerre polynomials, respectively.